An approximate method on non-equidistant partitions for double layer potential equation Original Research Article

A quadrature method for double layer potential equations with continuous coefficients on piecewise smooth curves is studied. The underlying grids are supposed to be locally non-equidistant. Such grids occur for instance in adaptive methods. Necessary and sufficient conditions for stability of the method are given in terms of invertibility of some well-defined model operators. These operators belong to an operator algebra for which the Fredholm properties of their elements are available. In particular, some bounds for indices of the mentioned operators are found. This fact includes a (weak) necessary condition for the invertibility.